Simple model of pileup for a quasi-monochromatic incident spectrum

The simple model starts by defining any x-ray event detected with the desired energy and grade to be ``good''. All other events (i.e., those with different energy or grade) are defined as ``bad''. Most practical x-ray line sources are not monochromatic, and are typically a mixture of a monochromatic line, other spectral features, and some continuum. The pileup model quantifies this aspect by defining that for every ``good'' x-ray event there are $\alpha$ ``bad'' events. Alternatively, if, in the absence of pileup, the number of good incident x-rays is N_i out of a total number of events N_T, then the fraction f_B of good x-ray events is $f_B=N_i/N_T=1/(1+\alpha)$. This model of a quasi-monochromatic beam is a good representation of the x-ray sources used in the ACIS quantum efficiency calibration.

Although for any CCD exposure the time history of event interactions is unknown, for the purposes of analysis we can picture the x-rays incident in one exposure as striking the CCD serially. The goal is to describe a function N_d(N_i) which represents the number of detected good x-rays as a function of the number of incident good x-ray events. By inverting this function, we can determine N_i from an experimental measurement of N_d. We begin construction of this function by examining the effect of a single x-ray.

Let $\epsilon$ be the effective area of the CCD affected by the occurrence of a ``good'' x-ray event (typically the desired energy is that of a K$_{\alpha }$ line the desired shapes are ASCA grades 0,2,3,4 and 6). In general, $\epsilon=\epsilon(E)$ will be a function of energy; the energy dependence is described in detail later. Similarly, $\epsilon ^{'}$ is the average effective area corresponding to any other x-rays, i.e. those with different energies and grades. The physical meaning of$\epsilon$ is that if a second x-ray is absorbed near a prior x-ray event such that the center of the second photoelectric absorption occurs within the area $\epsilon$ centered on the first x-ray, then an interaction occurs. Specific interaction effects are described mathematically below. We can derive the minimum size for $\epsilon$commensurate with our event detection criteria. All (standard) event discrimination is based on the 3x3 pixel subarray surrounding a local maximum of detected charge. Thus, any second x-ray landing within the subarray invokes an interaction, and the area of 9 pixels forms a lower limit for $\epsilon$. In all that follows, we express $\epsilon$ and $\epsilon ^{'}$ in units of the area of one quadrant of an ACIS CCID17 detector. In these units, a nine-pixel ``island'' has an area of $3.4\times10^{-5}$.
 
 

Figure 4.21: Regions of CCD for pileup model

The mathematical model begins by schematically dividing the area of a CCD as shown in Fig. 4.21. In our units, the total surface area of the detector (one quadrant of a CCID17) is normalized to unity. Let A₁ be the total area occupied by all good x-ray events (A₁ can also be interpreted as a probability or cross section for interaction; however the interpretation as an area is useful for developing a model). The number of detected events is taken to be $N_d =A_1/\epsilon$. This assumption is approximate since two good x-ray events could lie close enough together so that their $\epsilon$s overlap while they do not interact. Let A₂ be the total area occupied by charge produced by all other events. Then 1-A₁-A₂ is the CCD area unblemished by any interaction. The probability for an incident x-ray to land on a previous good x-ray event is A₁; the probability for an incident x-ray to land on a previous bad x-ray event is A₂, and the probability for an incident x-ray to land and be detected in unperturbed pixels is 1-A₁-A₂. The effect of an incident x-ray landing in A₁ is removal of one previous x-ray from A₁ while adding some area $\epsilon^{''}$ to A₂, which must be between 1 and 2 times $\epsilon$. We assume that any x-ray landing in A₂ does not change either A₁ or A₂. Finally, we assume that A₁ can increase only by good interactions in the unperturbed region 1-A₁-A₂. Then the change of A₁ and A₂ with per unit change in N_i can be described by the following pair of ordinary differential equations:
 
 

$$\frac{dA_1}{dN_T}=(1-A_1-A_2)f_B\epsilon - A_1\epsilon$$ (4.9)

 

$$\frac{dA_2}{dN_T}=(1-A_1-A_2)(1-f_B)\epsilon^{'} + A_1\epsilon^{''}$$ (4.10)

The solution is obtained by combining the two equations to separate variables. This results in a second order differential equation with the following solution,

$$A_1=\epsilon/\beta \exp(-\hat{\epsilon}N_i)\sin(\beta N_i )$$ (4.11)

or

$$N_d=f_B N_T e^{-\hat{\epsilon}f_B N_T}{\rm sinc}(f_B \beta N_T )$$ (4.12)

where

$$\hat{\epsilon}=\epsilon\left[1+\alpha(1+\epsilon^{'}/\epsilon)\right/2],$$ (4.13)

$$\beta = \sqrt{(1+\alpha)(\alpha\epsilon\epsilon^{'}+\epsilon\epsilon^{''})-\hat{\epsilon}^2}$$ (4.14)

and

$$A_1 = \epsilon N_d, \\f_B=N_i/N_T$$ (4.15)

Equation 4.12 has the desired asymptotic limit of N_i = N_d for low flux, with the following useful expansion for the logarithm of the ratio:

$$\log(\frac{N_d}{N_i})= - \hat{\epsilon}N_i -\beta^2N_i^2/6$$ (4.16)

In the limit where $\epsilon^{'} \sim \epsilon$ and $\epsilon^{''} \sim 3/2\epsilon$ then

$$\beta = \hat{\epsilon}/\sqrt{2(1+\alpha)}$$ (4.17)

Solutions to Eqn. 4.12 are plotted in Fig. 4.22 for two different flux ranges and for $\hat{\epsilon}$=0, 2, 4, ... 40 ($\times$10⁵) using $\epsilon^{''} =3/2\epsilon$. Specifically, the cross-section $\hat{\epsilon}$ equals the fractional area of a CCD region of interest, and N_d and N_i are the corresponding counts in that region. Note that $\hat{\epsilon}=10\times10^{-5}$ corresponds to a area of 26.2 pixels. The deviation from the line N_i = N_d increases as $\hat{\epsilon}$ increases. All the curves bend over for significantly high pileup and should assymtote to 0 for high enough N_i. This high pileup limit corresponds to severe charge cloud overlap so that very few x-rays satisfy the event selection criteria.

Figure 4.22:  Predicted relationship between detected and incident flux parameterized by the assumed value of  X 10⁵.  For example, the curve labeled 4 is the prediction for  = 4 X 10^-5 = 10.5  pixels.